A Fuzzy Based Solution for Improving Power Quality in Electric Railway Networks

There are many fundamental differences between electric traction networks and other industrial supply networks in terms of dynamic behavior and static characteristics. For example, the time variation of a load causes voltage variations in a supply network, which results in variations of power flow in the supply network. Today, reactive power compensators are the most practical solutions for keeping voltage levels in normal boundaries. In this paper, with the aim of fuzzy logic, a method for compensating reactive power losses in electrical traction networks is proposed. The proposed method has many advantages such as decreasing the reactive power compensation costs, determining the optimum switching step of capacitor banks and deducing the losses in electric traction networks.


Introduction
Reactive power creates numerous problems in electric railway supply networks.Because of the particular characteristics of electric traction loads, the problems have higher intensity in railway supply networks than other types of electric supply networks.Reactive power can be compensated by implementation of active or static methods.Nowadays a common method, in most electric substations, is to use shunt capacitors (Chin H. C., & Lin Wh. M., 1994;Baghzouz Y., & Ertem S., 1995).The advantages and disadvantages of this method are explained in Table 1.
Allocation of these shunt capacitors in the network shall be according to the power network topology.Electric railway networks has individual characteristic such as time variant and location variant trains' power demand.So, this paper proposed improving power quality based on optimization of capacitor allocation in electric railway power supply network.In next sections, this solution is expressed.

Calculation of the Capacitor Power for Power Factor Correction
In three phase circuits, the power factor correction capacitor is installed as shown in Figure 1.The capacitor power can be calculated from equation (1) (Chin Hong-Chan, & Lin Whei-min, 1996).
In Figure1, phases a and b are the supplying overhead lines in both sides and C 1 and C 2 are the shunt capacitors.
In a traction substation, the power factor is obtained from Where W P is the measured active power (kWh) and W Q is the measured reactive power (kvarh) (Baghzouz Y., & Ertem S., 1990).In the electric railway supply circuit, by considering the circuit, the capacitor power can be calculated from Where P o is the circuit no-load probability.Generally, implantation of capacitors in three phases (between the lag, lead or no-load phases) is not necessary.Capacitors are usually installed in circuits in the following ways (Shirmohamadi D., et al., 1988) In case II, it is also possible to install the capacitor in the lead phases; however, this would be harmful to the network power factor.

Cost Equation
After calculating the loss and considering the required capacitor for reactive power compensations, the cost value (Chin Hong-Chan, & Lin Whei-min, 1994) can be acquired from where K P is the cost of the reactive losses, P loss is the total losses of the overhead network in each step of capacitor allocation, J (=1,2,…,K) is the selected bus for capacitor installation, Kj is the minimum cost of possible circumstance coefficients plus the installation and maintenance, and Q j is the total power of the existing capacitors.

Problems' Constraints
Each optimization problem usually includes constraints.The operational constraint is the voltage amplitude in traction substations, which must be kept in the permitted limits of where V i is the voltage of substation i, and V max and V min are the maximum and minimum permitted voltages, respectively.

Determining the Required Capacitor Power via Fuzzy Logic
Various capacitor allocation methods are implemented for minimization of the reactive power losses.One of the implemented methods is based on Fuzzy logic that can assign sensitive points for capacitor installation.A Fuzzy-logic based solution benefits from high accuracy, response rate and μp (Chin Hong-Chan, & Lin Whei-min, 1996).
In this research, the active power losses are used as the static input data to the fuzzy sensitive system of voltage in the electric train location (or bus).
It should be noted that the minimum membership functions of the reactive power and bus voltage sensitiveness have assigned the required capacitor power in each operation period.Thus any increase in the losses results in an increase in the value of the membership function.Also, any decrease in the losses results in a decrease in the value of the membership function.One of the useful equations for demonstration of the function is where w is the weight coefficient obtained from where P(L i ) is the active power losses in the train to the substation circuit, P t is the total active power of other trains and Ρ Loss is the sum of the active power losses.
In order to find a solution for the bus voltage sensitiveness problem, in case of large voltage deviations, a low membership function is taken into account.Thus, the relative membership function can be expressed as where v(i) is the voltage of bus i, V max is the maximum voltage limit, V min is the minimum voltage limit and w is the weight coefficient acquired from equation 7. Generally for calculating the required capacitor power, both the bus voltage sensitiveness (as in Figure 2) and the active losses (as in Figure 3) factors should be considered.Therefore it becomes necessary to define a decision function that contains μs.
For finding the candidate locations for capacitor installation, both the active losses and the voltage sensitiveness factor are vital.Thus, considering the mentioned factors, a comprehensive decision function can be defined as The minimum point of the membership function is selected as the candidate location for capacitor installation.
The low value of μ s indicates that the point of the overhead network has a very high sensitiveness to voltage deviations and power losses.

Explanation of the Fuzzy-based Procedure
Figure 4 explains the Flowchart of the Optimum Capacitor Allocation Procedure in the Electric Railway Overhead Network.
The main steps of this procedure are (Ng H. N., et.al, 2000): i.
Calculating the network load flow, determining voltages and losses of the buses, and calculating μ p and μ v .
ii. Distinguishing the candidate point (i.e.bus M) by considering the minimum decision function.
iii.Determining the capacitor power, which satisfies both the voltage and the cost constraints, after performing the capacitor installation steps in bus M and running the load flow.If any capacitor power value does not satisfy voltage constraints of a selected bus, the bus should be eliminated from the list of candidate buses. iv.
After eliminating bus M from the decision function and setting the calculated capacitor on it, the load flow is run and a new decision function is formed without considering bus M.

v.
Distinguishing the new candidate bus from the new decision function, then jumping to step iii.
vi.The flowchart should be taken into account until it satisfies the voltage constraints.Also decreasing the cost value in an instant step from proceeding is lower than the defined minimum value for all buses.
One of the advantages of the demonstrated flowchart is that it considers the voltage constraints, while minimizing the active losses and therefore the corresponding cost function.In addition, the introduced procedure is capable of solving minimization of active losses problem (i.e.minimization of the cost function).The procedure can also determine the location and rate of the required capacitor for installation in the overhead network, provided that the bus voltages are in the permitted limits.

The Required Information
A radial supply feeder is considered and presented in Figure 5.The feeder has 9 load buses (an electric train per block), a nominal voltage of 25 kV and a rated power of 15 MVA.
The loads and the feeder data of the assumed overhead network are shown in Tables 2 and 3. Other data presented are: V min =0.9 p.u., V max =1.1 p.u. and kp =168 $/kW.
In Table 2, the maximum capacitor power should not exceed the total reactive power (i.e.4186 VAR).Therefore, regarding the existing capacitor and the minimum cost conditions (including the maintenance, installation and purchase costs), the capacitor step value and the corresponding cost coefficients can be acquired from Table 4.The network load flow results before compensation via the presented procedure are shown in Table 5.
It is clear from Table 6 that the costs and total losses in the presented procedure have more appropriate values than the referenced value in (Chin H. C., & Lin Wh. M., 1994).

Flowchart Implementation Results
According to Table 7, the probable rates of the capacitors are obtained by considering the corresponding cost coefficients in every step of the calculations.The voltage magnitudes for each selected item are shown in Table 8.

Conclusion
An appropriate method for determining the capacitors rates in each step, by considering the voltage magnitude constraints and the cost of losses, was proposed.In the proposed method, by setting the capacitor conditions in each period of operation and considering the absorbed power from each substation and the number of trains in each line, it became possible to ensure the improvement of the power quality index of the electric railway networks.In addition, the proposed Fuzzy-based solution had many advantages such as optimization of the compensation steps and reduction of the compensation costs.Shirmohamadi, D., et al. (1988)

Figure 1 .
Figure 1.Power factor correction in a three-phase circuit by using shunt capacitors

Table 6 .
(Chin H. C., 1994)paring the presented procedure with reference value Reference(Chin H. C., 1994)Defined method in this paper

Table 7 .
Variable choices of capacitor rates and the corresponding cost coefficients

Table 8 .
The voltages of the bus after capacitor placement